14.5 Duality Results and Maximal Elements 315Claim 1 can now be improved as follows: for−∞<β≤^18 we haveE[exp(βν)] = 2
1 −√ 1 − 8 β(^2).
As above let us now look atσ=inf{t|Wt−^12 t≥ln 2}and consider
the measure defined byddQP=Zσ=2 (^1) {σ<∞}. The extension of Girsanov’s
theorem for non-equivalent measures, due E. Lenglart [L 77], allows to write
the Brownian motionWasWt=Wt′+t,whereW′is aQ-Brownian motion
and the equations holdQ-a.s.. It follows that for 0≤β≤^18
EQ[exp(βσ)] = 2E[exp(βσ) (^1) {σ<∞}]=E[exp(βν)] = 2
1 −√ 21 − 8 β
.
In other wordsE[exp(βσ) (^1) {σ<∞}]=2
− 1 −√ 21 − 8 β
.
We are now ready to show:
Claim 3: EQ 0 [Lγτ∧σ]<∞for 1≤γ≤1+
√
2
2.
The calculations are straightforward but we prefer to give the details
E[Lγτ∧σZτ∧σ]
=E
[
exp(
γBτ∧σ−
γ
2τ∧σ)
exp(
Wτ∧σ−1
2
τ∧σ)]
=E
[
exp(
γBτ∧σ−γ^2
2
τ∧σ)
exp(
γ(γ−1)
2
τ∧σ)
exp(
Wτ∧σ−1
2
τ∧σ)]
≤E
[
exp(
γBτ∧σ−γ^2
2τ∧σ)
exp(
γ(γ−1)
2σ)
exp(
Wσ−1
2
σ)]
≤ lim
n→∞E
[
exp(
γBτ∧σ∧n−
γ^2
2τ∧σ∧n)
exp(
γ(γ−1)
2σ)
exp(
Wσ−1
2
σ)]
≤nlim→∞E[
exp(
γBτ∧n−γ^2
2
τ∧n)
exp(
γ(γ−1)
2
σ)
exp(
Wσ−1
2
σ)]
,
this is seen as follows: we split into the two sets{τ∧n≤σ}and{τ∧n
>σ}∈Fσand use the martingale property of exp(Bt−^12 t)fort≤n.≤E[
exp(
γ(γ−1)
2σ)
exp(
Wσ−1
2
σ)]
by independence ofBandW!For 1≤γ≤1+
√
2
2 we haveγ(γ−1)
2 ≤1
8 and henceEQ 0 [Lγτ∧σ]≤E[
exp(
γ(γ−1)
2σ)
(^21) {σ<∞}
]
<∞.
This ends the proof of (iv) and completes the discussion of the example.
We now turn to the characterisation of maximal and of attainable elements.
The approach is different from the one used in Chap. 11, which was based on
a change of num ́eraire technique. In order not to overload the statements we
henceforth suppose thatwis a feasible weight function.